Truncation Errors in the Stokes and Vening Meinesz Formulae for Different Order Spherical Harmonic Gravity Terms

Abstract

A detailed analysis of truncation errors in the Stokes formula integration, using Molodenskii's method, shows the mode of dependence of the errors on the spherical harmonic components of Ag of different order. The results indicate that significant reduction in the truncation errors can be achieved by adopting a reference model for normal gravity of higher order than that based on the International Ellipsoid. Particularly, the use of a seventh order reference model combined with truncation at the first zero crossing of the Stokes kernel function appears very promising. The treatment of truncation errors for deflection of the vertical as given by Molodenskii et al. and Hirvonen & Moritz yields results for deflections as derivatives of geoidal heights explicitly obtained from Stokes formula. These errors can be reduced by the same techniques as suggested for the Stokes integration. The truncation error behaviour of the Vening Meinesz formulae is shown to be different. An adaptation of Molodenskii's approach using an expansion of the truncated deflection of the vertical kernel function in terms of associated Legendre polynomials of first order provides a means for spherical harmonic analysis of the truncation errors in the Vening Meinesz formulae. 449 1

Accuracy of Geoid Heights from Modified Stokes Kernels

“…The lack of gravity measurements over much of the Earth's surface is still the major problem in gravimetric geodesy. The combination of satellite derived spherical harmonics with gravimetry in order to reduce the latter requirement to a localized region for geoid height and deflection of the vertical calculations has been considered by DeWitte (1967) and Rapp (1967). In both references, the classical Stokes kernel was used in connection with measured gravity anomalies to calculate geoid height.…”

Generalized geoidal estimators for deterministic modifications of spherical Stokes' function

“…Using the properties of orthogonal series expansions more rapid convergence for the amplitudes of the truncation error coefficients and consequently the truncation error is achieved without the direct criterion of minimization. Reduced magnitude of the truncation error near zeros of spherical Stokes' function was observed by de Witte (1967). Meissl (1971) proposed an integration kernel in the form of algebraic subtraction of spherical Stokes' function and its value at integration radius.…”

Utilisation of Fast Fourier Transform and Least-squares Modification of Stokes formula to compile a gravimetric geoid model over Khartoum State: Sudan

“…Some authors employed the RCR approach using high degree coefficients of the GGM for generating a higher degree reference field and the residual field is computed from the integral formula (see e.g. Jeffreys, 1953;De Witte, 1967;Wong and Gore, 1969). More studies of the deterministic modifications to Stokes functions based on the RCR approach were discussed and investigated by Featherstone et al (1998); Vaníček and Featherstone (1998);Featherstone (2003).…”